Page 354 - DMTH504_DIFFERENTIAL_AND_INTEGRAL_EQUATION

Unit 21: Classifications of Second Order Partial Differential Equations

We find that the most general form of the partial differential equation of the second order will Notes
be of the form
F(x, y, z, p, q, r, s, t) = 0 ...(2)

Example: Consider z as a function of x, y through two functions f and g as follows
2
2
z = f(x y) + g(x + y)= 0 ...(3)
Find the differential equation by eliminating f and g
Solution:

z f q
p =
x x x
2
2
Let u = x y and v = x + y, so that
z = f(u) + g(v)

z f u q v
then p = . .
x u x v x
f q f q
x
x
= .(2 ) (2 ) 2x ...(4)
u v u v
f f
q = ( 1) .(1)
u v
f f
= ...(5)
u v
2 z f q 2 f 2 f
r = 2 p = 2 2x 2x 2x
x x u v u 2 v 2

f q 2 2 f 2 f
= 2 4x ...(6)
u v u 2 v 2
2 2 2
z f u f v
q =
x y x u 2 x v 2 x
2 f 2 f
= 2x 2 2x 2 ...(7)
x v
2
z 2 f u 2 f v
q = .
y 2 y u 2 y v 2 y
2 f 2 f
= 2 2 ...(8)
u v
Now using equations (4), (6) and (8) we have
2 z f q 2 f 2 f
r = 2 = 2 4x 2 2 2
x u v u v

2 z 1 z 2 z
or 2 = 4x 2 2 ...(9)
x x x y

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